Hi Giorgios,
the issue is not so much the LCIA method but the "inventory", i.e. the social indicator values in the LCA model. This is not provided in ecoinvent (original), but contained in the soca database that we created as add-on to ecoinvent. Some IO databases like exiobase have some few social indicators and can provide these, but the indicator set is quite limited compared to the SHDB or PSILCA database.
Best wishes,
Andreas